Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 9 Hypothesis Testing Developing Null and Alternative Hypotheses Type I and Type II Errors Population Mean: s Known Population Mean: s Unknown Population Proportion © 2006 Thomson/South-Western Slide 1 Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected. The null hypothesis, denoted by H0 , is a tentative assumption about a population parameter. The alternative hypothesis, denoted by Ha, is the opposite of what is stated in the null hypothesis. The alternative hypothesis is usually what the test is attempting to establish. © 2006 Thomson/South-Western Slide 2 Developing Null and Alternative Hypotheses Testing in Decision-Making Situations • A decision maker might have to choose between two courses of action, one associated with the null hypothesis and another associated with the alternative hypothesis. • Example: Accepting a shipment of goods from a supplier or returning the shipment of goods to the supplier © 2006 Thomson/South-Western Slide 3 Steps of Hypothesis Testing Step 1. Develop the null and alternative hypotheses. Step 2. Specify what is expected under the null. Select a . Step 3. Collect the sample data and compute the test statistic to compare observed and expected results. Step 4. Use the value of the test statistic to compute the p-value OR choose the critical value Step 5. Reject H0 if p-value < a (chosen sig. level) OR Reject H0 if test stat. > critical value © 2006 Thomson/South-Western Slide 4 Summary of Forms for Null and Alternative Hypotheses about a Population Mean The equality part of the hypotheses always appears in the null hypothesis. In general, a hypothesis test about the value of a population mean must take one of the following three forms (where 0 is the hypothesized value of the population mean). H 0 : 0 H a : 0 H 0 : 0 H a : 0 H 0 : 0 H a : 0 One-tailed (lower-tail) One-tailed (upper-tail) Two-tailed © 2006 Thomson/South-Western Slide 5 Null and Alternative Hypotheses Example: Metro EMS A major west coast city provides one of the most comprehensive emergency medical services in the world. Operating in a multiple hospital system with approximately 20 mobile medical units, the service goal is to respond to medical emergencies with a mean time of 12 minutes or less. © 2006 Thomson/South-Western Slide 6 One-Tailed Tests About a Population Mean: s Known Example: Metro EMS The response times for a random sample of 40 medical emergencies were tabulated. The sample mean is 13.25 minutes. The population standard deviation is believed to be 3.2 minutes. The EMS director wants to perform a hypothesis test, with a .05 level of significance, to determine whether the service goal of 12 minutes or less is being achieved. © 2006 Thomson/South-Western Slide 7 Null and Alternative Hypotheses H0: The emergency service is meeting the response goal; no follow-up action is necessary. Ha: The emergency service is not meeting the response goal; appropriate follow-up action is necessary. where: = mean response time for the population of medical emergency requests © 2006 Thomson/South-Western Slide 8 One-Tailed Tests About a Population Mean: s Known p -Value and Critical Value Approaches 1. Develop the hypotheses. H0: Ha: 2. Specify the level of significance. a = .05 3. Compute the value of the test statistic. x 13.25 12 z 2.47 s / n 3.2 / 40 © 2006 Thomson/South-Western Slide 9 Critical Value Approach to One-Tailed Hypothesis Testing The test statistic z has a standard normal probability distribution. We can use the standard normal probability distribution table to find the z-value with an area of a in the lower (or upper) tail of the distribution. The value of the test statistic that established the boundary of the rejection region is called the critical value for the test. The rejection rule is: • Lower tail: Reject H0 if z < -za • Upper tail: Reject H0 if z > za © 2006 Thomson/South-Western Slide 10 One-Tailed Tests About a Population Mean: s Known Critical Value Approach 4. Determine the critical value and rejection rule. For a = .05, z.05 = 1.645 Reject H0 if z > 1.645 5. Determine whether to reject H0. Because 2.47 > 1.645, we reject H0. We are at least 95% confident that Metro EMS is not meeting the response goal of 12 minutes. © 2006 Thomson/South-Western Slide 11 Upper-Tailed Test About a Population Mean: s Known Critical Value Approach Sampling distribution of z x 0 s/ n Reject H0 Do Not Reject H0 a z 0 © 2006 Thomson/South-Western za = 1.645 Slide 12 p-Value Approach to One-Tailed Hypothesis Testing The p-value is the probability, computed using the test statistic, that measures the support (or lack of support) provided by the sample for the null hypothesis. If the p-value is less than or equal to the level of significance a, the value of the test statistic is in the rejection region. Reject H0 if the p-value < a . © 2006 Thomson/South-Western Slide 13 One-Tailed Tests About a Population Mean: s Known p –Value Approach 4. Compute the p –value. For z = 2.47, cumulative probability = .9932. p–value = 1 .9932 = .0068 5. Determine whether to reject H0. Because p–value = .0068 < a = .05, we reject H0. We are at least 95% confident that Metro EMS is not meeting the response goal of 12 minutes. © 2006 Thomson/South-Western Slide 14 One-Tailed Tests About a Population Mean: s Known p –Value Approach Sampling distribution of z x 0 s/ n a = .05 p-value z 0 © 2006 Thomson/South-Western za = 1.645 z= 2.47 Slide 15 Two-Tailed Tests About a Population Mean: s Known p –Value and Critical Value Approaches 1. Determine the hypotheses. H0: Ha: 6 2. Specify the level of significance. a = .03 3. Compute the value of the test statistic. x 0 6.1 6 z 2.74 s / n .2 / 30 © 2006 Thomson/South-Western Slide 16 Example: Glow Toothpaste Two-Tailed Test About a Population Mean: s Known The production line for Glow toothpaste is designed to fill tubes with a mean weight of 6 oz. Periodically, a sample of 30 tubes will be selected in order to check the filling process. Quality assurance procedures call for the continuation of the filling process if the sample results are consistent with the assumption that the mean filling weight for the population of toothpaste tubes is 6 oz.; otherwise the process will be adjusted. © 2006 Thomson/South-Western Slide 17 Example: Glow Toothpaste Two-Tailed Test About a Population Mean: s Known Assume that a sample of 30 toothpaste tubes provides a sample mean of 6.1 oz. The population standard deviation is believed to be 0.2 oz. Perform a hypothesis test, at the .03 level of significance, to help determine whether the filling process should continue operating or be stopped and corrected. © 2006 Thomson/South-Western Slide 18 Critical Value Approach to Two-Tailed Hypothesis Testing The critical values will occur in both the lower and upper tails of the standard normal curve. Use the standard normal probability distribution table to find za/2 (the z-value with an area of a/2 in the upper tail of the distribution). The rejection rule is: Reject H0 if z < -za/2 or z > za/2. © 2006 Thomson/South-Western Slide 19 Two-Tailed Tests About a Population Mean: s Known Critical Value Approach 4. Determine the critical value and rejection rule. For a/2 = .03/2 = .015, z.015 = 2.17 Reject H0 if z < -2.17 or z > 2.17 5. Determine whether to reject H0. Because 2.47 > 2.17, we reject H0. We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz. © 2006 Thomson/South-Western Slide 20 Two-Tailed Tests About a Population Mean: s Known Critical Value Approach Sampling distribution x 0 of z s/ n Reject H0 Reject H0 Do Not Reject H0 a/2 = .015 -2.17 © 2006 Thomson/South-Western a/2 = .015 0 2.17 z Slide 21 p-Value Approach to Two-Tailed Hypothesis Testing Compute the p-value using the following three steps: 1. Compute the value of the test statistic z. 2. If z is in the upper tail (z > 0), find the area under the standard normal curve to the right of z. If z is in the lower tail (z < 0), find the area under the standard normal curve to the left of z. 3. Double the tail area obtained in step 2 to obtain the p –value. The rejection rule: Reject H0 if the p-value < a . © 2006 Thomson/South-Western Slide 22 Two-Tailed Tests About a Population Mean: s Known p –Value Approach 4. Compute the p –value. For z = 2.74, cumulative probability = .9969 p–value = 2(1 .9969) = .0062 5. Determine whether to reject H0. Because p–value = .0062 < a = .03, we reject H0. We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz. © 2006 Thomson/South-Western Slide 23 Two-Tailed Tests About a Population Mean: s Known p-Value Approach 1/2 p -value = .0031 1/2 p -value = .0031 a/2 = a/2 = .015 .015 z z = -2.74 -za/2 = -2.17 © 2006 Thomson/South-Western 0 za/2 = 2.17 z = 2.74 Slide 24 Type I Error Because hypothesis tests are based on sample data, we must allow for the possibility of errors. A Type I error is rejecting H0 when it is true. The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance. Applications of hypothesis testing that only control the Type I error are often called significance tests. © 2006 Thomson/South-Western Slide 25 Type II Error A Type II error is accepting H0 when it is false. It is difficult to control for the probability of making a Type II error. Statisticians avoid the risk of making a Type II error by using “do not reject H0” and not “accept H0”. © 2006 Thomson/South-Western Slide 26 Type I and Type II Errors Population Condition Conclusion H0 True ( < 12) H0 False ( > 12) Accept H0 (Conclude < 12) Correct Decision Type II Error Type I Error Correct Decision Reject H0 (Conclude > 12) © 2006 Thomson/South-Western Slide 27 Tests About a Population Mean: s Unknown Test Statistic x 0 t s/ n This test statistic has a t distribution with n - 1 degrees of freedom. © 2006 Thomson/South-Western Slide 28 Tests About a Population Mean: s Unknown Rejection Rule: p -Value Approach Reject H0 if p –value < a Rejection Rule: Critical Value Approach H0: Reject H0 if t < -ta H0: Reject H0 if t > ta H0: Reject H0 if t < - ta or t > ta © 2006 Thomson/South-Western Slide 29 p -Values and the t Distribution The format of the t distribution table provided in most statistics textbooks does not have sufficient detail to determine the exact p-value for a hypothesis test. However, we can still use the t distribution table to identify a range for the p-value. An advantage of computer software packages is that the computer output will provide the p-value for the t distribution. © 2006 Thomson/South-Western Slide 30 Example: Highway Patrol One-Tailed Test About a Population Mean: s Unknown A State Highway Patrol periodically samples vehicle speeds at various locations on a particular roadway. The sample of vehicle speeds is used to test the hypothesis H0: < 65 The locations where H0 is rejected are deemed the best locations for radar traps. © 2006 Thomson/South-Western Slide 31 Example: Highway Patrol One-Tailed Test About a Population Mean: s Unknown At Location F, a sample of 64 vehicles shows a mean speed of 66.2 mph with a standard deviation of 4.2 mph. Use a = .05 to test the hypothesis. © 2006 Thomson/South-Western Slide 32 One-Tailed Test About a Population Mean: s Unknown p –Value and Critical Value Approaches 1. Determine the hypotheses. H0: < 65 Ha: > 65 2. Specify the level of significance. a = .05 3. Compute the value of the test statistic. t x 0 66.2 65 2.286 s / n 4.2 / 64 © 2006 Thomson/South-Western Slide 33 One-Tailed Test About a Population Mean: s Unknown Critical Value Approach 4. Determine the critical value and rejection rule. For a = .05 and d.f. = 64 – 1 = 63, t.05 = 1.669 Reject H0 if t > 1.669 5. Determine whether to reject H0. Because 2.286 > 1.669, we reject H0. We are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph. Location F is a good candidate for a radar trap. © 2006 Thomson/South-Western Slide 34 One-Tailed Test About a Population Mean: s Unknown Reject H0 Do Not Reject H0 0 © 2006 Thomson/South-Western a ta = 1.669 t Slide 35 One-Tailed Test About a Population Mean: s Unknown p –Value Approach 4. Compute the p –value. For t = 2.286, the p–value must be less than .025 (for t = 1.998) and greater than .01 (for t = 2.387). .01 < p–value < .025 5. Determine whether to reject H0. Because p–value < a = .05, we reject H0. We are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph. © 2006 Thomson/South-Western Slide 36 A Summary of Forms for Null and Alternative Hypotheses About a Population Proportion The equality part of the hypotheses always appears in the null hypothesis. In general, a hypothesis test about the value of a population proportion p must take one of the following three forms (where p0 is the hypothesized value of the population proportion). H 0 : p p0 H a : p p0 H 0 : p p0 H a : p p0 H 0 : p p0 H a : p p0 One-tailed (lower tail) One-tailed (upper tail) Two-tailed © 2006 Thomson/South-Western Slide 37 Tests About a Population Proportion Test Statistic z p p0 sp where: sp p0 (1 p0 ) n assuming np > 5 and n(1 – p) > 5 © 2006 Thomson/South-Western Slide 38 Tests About a Population Proportion Rejection Rule: p –Value Approach Reject H0 if p –value < a Rejection Rule: Critical Value Approach H0: p p Reject H0 if z > za H0: p p Reject H0 if z < -za H0: pp Reject H0 if z < -za or z > za © 2006 Thomson/South-Western Slide 39 Two-Tailed Test About a Population Proportion Example: National Safety Council For a Christmas and New Year’s week, the National Safety Council estimated that 500 people would be killed and 25,000 injured on the nation’s roads. The NSC claimed that 50% of the accidents would be caused by drunk driving. © 2006 Thomson/South-Western Slide 40 Two-Tailed Test About a Population Proportion Example: National Safety Council A sample of 120 accidents showed that 67 were caused by drunk driving. Use these data to test the NSC’s claim with a = .05. © 2006 Thomson/South-Western Slide 41 Two-Tailed Test About a Population Proportion p –Value and Critical Value Approaches 1. Determine the hypotheses. H 0 : p .5 H a : p .5 2. Specify the level of significance. a = .05 3. Compute the value of the test statistic. a common error is using p in this formula p0 (1 p0 ) .5(1 .5) sp .045644 n 120 z p p0 sp © 2006 Thomson/South-Western (67 /120) .5 1.28 .045644 Slide 42 Two-Tailed Test About a Population Proportion pValue Approach 4. Compute the p -value. For z = 1.28, cumulative probability = .8997 p–value = 2(1 .8997) = .2006 5. Determine whether to reject H0. Because p–value = .2006 > a = .05, we cannot reject H0. © 2006 Thomson/South-Western Slide 43 Two-Tailed Test About a Population Proportion Critical Value Approach 4. Determine the criticals value and rejection rule. For a/2 = .05/2 = .025, z.025 = 1.96 Reject H0 if z < -1.96 or z > 1.96 5. Determine whether to reject H0. Because 1.278 > -1.96 and < 1.96, we cannot reject H0. © 2006 Thomson/South-Western Slide 44